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Theorem raleqbidvv 3332
Description: Version of raleqbidv 3344 with additional disjoint variable conditions, not requiring ax-8 2108 nor df-clel 2814. (Contributed by BJ, 22-Sep-2024.)
Hypotheses
Ref Expression
raleqbidvv.1 (𝜑𝐴 = 𝐵)
raleqbidvv.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
raleqbidvv (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem raleqbidvv
StepHypRef Expression
1 raleqbidvv.1 . 2 (𝜑𝐴 = 𝐵)
2 raleqbidvv.2 . . 3 (𝜑 → (𝜓𝜒))
32adantr 480 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
41, 3raleqbidva 3330 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  wral 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-ral 3060  df-rex 3069
This theorem is referenced by:  rexeqbidvvOLD  3335  raleqbi1dv  3336  gpg5nbgrvtx03star  47971  postcposALT  48884
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